On-line and Approximation Algorithms Fall Semester, 2004/2005 Exercise 4 (EXAM): January 12, 2005 Summary: On-line and Approximation Algorithms Fall Semester, 2004/2005 Exercise 4 (EXAM): January 12, 2005 Lecturer: Prof. Yossi Azar Write short but full and accurate answers. Each question should start on a new page and should not exceed a page (both sides). 1. We are given a connected graph G = (V, E), where |V | = n. All edges have unit capacity. At step i we receive a request (si, ti) of bandwidth precisely 1/3. The goal is to maximize the total bandwidth of accepted requests (preemption is not allowed). (a) Design an O(n1/3) competitive algorithm for the problem. Hint: use weights of about 0, n1/3, n2/3, n. (b) How would you modify the algorithm and the analysis to achieve an O(kn1/3) competitive algorithm if all requests are of bandwidth between 1/k and 1/3 for any k > 3. 2. Consider on-line load balancing of permanent tasks on m machines in the restricted assignment model. Consider the case where for each i the set of machines which is associated with job i is [1 . . . ki] for some 1 ki m. (a) Design a constant competitive algorithm. (b) Show a lower bound of 2 for deterministic algorithms, for any m 6, using only unit jobs. (c) Show a lower bound of 2 - O(1/m) for randomized algorithms, for any m, using only unit jobs. 3. Suppose we are given one machine and a set of jobs that arrive over time. The machine can Collections: Computer Technologies and Information Sciences